\begin{itemize}
\item
To find the approximate range for $\sigma$, $\vartheta$, and $L$, we have
evaluated the mean squared error (MSE) with every combination from ${0.01, 0.1,
1, 10, 100, 100}$ for the variables.
The training set consisted of the first 10 examples, and the evaluation was
done on the remaining 5.
The lowest MSE was $99.812$ for $\sigma=0.1 , \vartheta=1, L=10$.

\item
The MSE varied for different sizes of the test set, but was only significantly
worse for one or two training examples. For set sizes between 3 and 10, the MSE
was between 74.946 for a size of 7 examples, and 121.18 for 6 data points.

Due to the small set of data it is hard to tell what the reason is. One
possibility is that the test set was for a large part in a section that was
sparse in training data, or that the data was so noisy that three data points
were enough to give an accurate prediction.

\item
The variance is high where the data is sparse, and near 0 where it is more
dense. Surprisingly, the variance is not higher where the variance of the data
is higher, but it is low in a smaller region, as can be seen in figure
\ref{fig:variance}.
\begin{figure}
	\centering
	\includegraphics[scale=0.4]{resources09/variance}
	\caption{Variance of the function and data points of the training set}
	\label{fig:variance}
\end{figure}

\item
	A smaller $L$ (figure \ref{fig:l01}) makes the function match the points in the training set more
	closely, where a larger $L$ (figure \ref{fig:l10}) describes a simpler function that fits the more
	genaral trends of the data. For small $L$s the function remains flat where
	the data is sparse, while bigger values for $\vartheta$ also fit the points more
	closely, but deviate farther from the mean where data is sparse.

\begin{figure}
	\centering
	\includegraphics[scale=0.4]{resources09/l10}
	\caption{$L=10$}
	\label{fig:l10}
\end{figure}

\begin{figure}
	\centering
	\includegraphics[scale=0.4]{resources09/l01}
	\caption{$L=0.1$}
	\label{fig:l01}
\end{figure}

Increasing $\sigma^2$ makes the function more flat overall, and approaches a
line on the mean as it goes to infinite.
\end{itemize}
